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Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups

Yeeka Yau

TL;DR

This work generalizes Reading's canonical join representation from finite to all finitely generated Coxeter groups, linking inversion geometry to the cone-type framework. It proves that for every $w\in W$ and each $\\beta \in \\Phi^R(w)$ there is a unique minimal $j_{\\beta}$ with $\\beta \\in \\Phi(j_{\\beta})$, giving $w = \\bigvee_{\\beta \\in \\Phi^R(w)} j_{\\beta}$, and clarifies how boundary-root witnesses govern cone-type structure via tight gates. The authors introduce the $\\mathscr{T}^0$ partition and show that $\\Gamma^0$, the tight gates, are exactly the join-irreducible elements of $\\Gamma$, and that cone types are determined by these gates; this yields efficient methods to test cone-type equivalence by computing a smaller core set. They propose practical algorithms to compute the super-elementary roots and tight gates using elementary roots and low elements, together with suffix-closure properties, and provide data for select Coxeter groups to demonstrate scalability. Overall, the results illuminate the interplay between inversion sets, cone types, and gate structures, offering both a uniform theoretical framework and concrete computational tools for Coxeter groups and their automata-theoretic representations.

Abstract

In this article, we establish some new combinatorial properties of elements in Coxeter groups. Firstly, we generalise Reading's theorem on the canonical join representations of elements in finite Coxeter groups to all finitely generated Coxeter groups. Secondly, we show that for any element $x$ in a Coxeter group $W$ and root $β$ in its inversion set $Φ(x)$, the set of elements $y \in W$ satisfying $Φ(x) \cap Φ(y) = \{ β\} $ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.

Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups

TL;DR

This work generalizes Reading's canonical join representation from finite to all finitely generated Coxeter groups, linking inversion geometry to the cone-type framework. It proves that for every and each there is a unique minimal with , giving , and clarifies how boundary-root witnesses govern cone-type structure via tight gates. The authors introduce the partition and show that , the tight gates, are exactly the join-irreducible elements of , and that cone types are determined by these gates; this yields efficient methods to test cone-type equivalence by computing a smaller core set. They propose practical algorithms to compute the super-elementary roots and tight gates using elementary roots and low elements, together with suffix-closure properties, and provide data for select Coxeter groups to demonstrate scalability. Overall, the results illuminate the interplay between inversion sets, cone types, and gate structures, offering both a uniform theoretical framework and concrete computational tools for Coxeter groups and their automata-theoretic representations.

Abstract

In this article, we establish some new combinatorial properties of elements in Coxeter groups. Firstly, we generalise Reading's theorem on the canonical join representations of elements in finite Coxeter groups to all finitely generated Coxeter groups. Secondly, we show that for any element in a Coxeter group and root in its inversion set , the set of elements satisfying is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of and leads to efficient computational methods to determine whether arbitrary elements of have the same cone type.

Paper Structure

This paper contains 13 sections, 39 theorems, 55 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter Groups
  4. Standard Geometric Representation and root system of $W$
  5. The Coxeter complex
  6. The Cayley graph of $(W,S)$
  7. Cone Types and Boundary roots of $W$
  8. Maximal dihedral reflection subgroups and the short inversion poset
  9. Canonical join-representations
  10. Witnesses of boundary roots
  11. The $\mathscr{T}^0$ partition and join-irreducible elements of $\Gamma$
  12. Computing the sets $\mathcal{S}$ and $\Gamma^0$
  13. Data for select Coxeter groups

Key Result

Theorem 1

GratzerWehrung2016 Suppose $W$ is a finite Coxeter group and $w \in W$. For each $\beta \in \Phi^R(w)$ there is a unique minimal length element $j_{\beta} \in \{ v \mid v \preceq w, \beta \in \Phi(v) \}$. The canonical join representation of $w$ is $w = \bigvee \{ j_{\beta} \mid \beta \in \Phi^R(w)

Figures (11)

  • Figure 1: Example of Reading's canonical join representation theorem from the Coxeter group of type $A_2$.
  • Figure 2: Illustration of \ref{['thm:canonical_join_representation']}
  • Figure 3: Example of a cone type in the Coxeter group of type $\widetilde{G}_2$.
  • Figure 4: Let $x$ and $y$ be the elements illustrated above. One can see that $T(x^{-1}) = T(y^{-1})$, so $x$ and $y$ are in the same cone type part $Q(T)$; which is the region bounded by the dashed lines, and the horizontal black and blue hyperplanes (the region looks like a downward right pointing arrow). The cone type $T:= T(x^{-1}) = T(y^{-1})$ is represented by the gray shaded region. Let $\beta \in \partial T$ correspond to the blue hyperplane. The region bounded by the red highlighted lines is $\partial T(y^{-1})_{\beta}$ and the region bounded by the blue highlighted lines is $\partial T(x^{-1})_{\beta}$.
  • Figure 5: Let the element $x$ be indicated as above. The grey shaded region is the cone type $T:=T(x^{-1})$ (the darker grey alcove represents the identity) with boundary roots $\partial T$ indicated by the thickened black walls. The yellow shaded region is the cone type part $Q(T)$. The remaining coloured shaded regions are the sets $\partial T(x^{-1})_{\beta}$ for each $\beta \in \partial T$ and the corresponding coloured dots are the gates of those regions.
  • ...and 6 more figures

Theorems & Definitions (74)

  • Definition 1
  • Theorem 1
  • Example 1.1
  • Theorem 1
  • Example 1.2
  • Theorem 2
  • Example 1.3
  • Definition 1.4
  • Remark 1.1
  • Definition 1.5
  • ...and 64 more